BANGALORE, India—As its growth slows, India’s technology services outsourcing industry is struggling to become less dependent on peddling the services of inexpensive programmers and trying to enter new businesses like off-the-shelf software and helping customers with big data and cloud computing.

India’s leading outsourcers, including Tata Consultancy Services Ltd. and Infosys Ltd., said last month that their profit growth was squeezed last quarter by dwindling demand from the U.S. and Europe.

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